On these grounds,
the Hermitization
did not proceed directly
from \({\cal H}_{math}\)  to \({\cal H}_{phys}\) 
(as, for example, in Ref. [1]}) but rather
indirectly,
via the third inner-product
space (i.e., Hilbert or Krein space) \({\cal H}_{intermediate}\) .
A key to the comparison
of ISP and PTQM
has been found in the charge.
Indeed, in
the PTQM framework
the theory only becomes consistent
after one guarantees the \({\cal PCT}-\) symmetry
\(\,H{\cal PCT}={\cal PCT}H\,\) 
of the Hamiltonian [2]}.
In parallel, in the ISP context
the charge appeared to play
a double role, i.e., not only the original role of
component of the correct physical inner-product metric
(REF ) which determines
the geometry in \({\cal H}_{phys}={\cal R}_0\) , but also
a new role of an auxiliary metric operator
in
\({\cal H}_{intermediate}={\cal R}_1\) .
